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Unit 4 · 1.4

Five-number summary & box plots

Median-excluded quartiles, IQR, 1.5·IQR outlier limits, reading box plots

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Rule cards

Quartiles: leave the median OUT

Sort first. Q1 = middle of the values BEFORE the median, Q3 = middle of the values AFTER it; when n is odd the median is in neither half.

  1. Sort the data (stat → 2:SortA().
  2. Median = middle value (average the two middles if n is even).
  3. n odd → drop the median; n even → split straight down the middle.
  4. Q1 = median of the lower half; Q3 = median of the upper half.
  5. Check against 1-Var Stats: it uses the same rule.
Traps
  • Put the median into both halves.
    14.6, 15.5, 16.3 | 16.3, 18.2, 19.8 → Q1 = 15.5, Q3 = 18.2
    ✓ halves 14.6, 15.5 and 18.2, 19.8 → Q1 = 15.05, Q3 = 19 (this is the key's error on HW 1.4 #6)
  • Found the quartiles on the unsorted list.
    read the 2nd and 4th values as typed
    ✓ SORT before anything
  • Took the value next to the middle.
    sunflowers n = 23 → 13th value 28
    ✓ median = 12th value = 27
  • Used the wrong position inside a half.
    upper 11 values → 7th = 41
    ✓ upper 11 → 6th = Q3 = 40
TI-84 Evo keys

IQR and outlier limits

IQR = Q3 − Q1. Lower limit = Q1 − 1.5·IQR, upper limit = Q3 + 1.5·IQR; an outlier is below the lower or above the upper limit.

  1. Find Q1 and Q3 (median excluded).
  2. IQR = Q3 − Q1.
  3. Multiply: 1.5 × IQR.
  4. Lower limit = Q1 − that; upper limit = Q3 + that.
  5. Any data value outside the limits is an outlier.
Traps
  • Used IQR instead of 1.5·IQR.
    20 − 4 and 20.5 + 4
    ✓ 1.5 × 4 = 6 → 16.5 − 6 = 10.5 and 20.5 + 6 = 26.5
  • Used max − min.
    19.8 − 14.6 = 5.2
    ✓ IQR = Q3 − Q1
  • Called the min or max an outlier without checking the limits.
    71 is the biggest → outlier
    ✓ 71 is an outlier only because 71 > 70 (40 + 30)

Reading a box plot

Left whisker end = Min, box left edge = Q1, line inside = median, right edge = Q3, right whisker end = Max; each section holds 25%.

Traps
  • Read the top 25% from the wrong side.
    top 25% = Min to Q1
    ✓ top 25% = Q3 to Max (sunflowers: 40 to 71)
  • Drew the box from Min to Max.
    box spans the whole data
    ✓ box = Q1 to Q3, the middle 50%
  • Took the middle of the box as the median.
    median = (Q1 + Q3) ÷ 2
    ✓ read the line inside the box

Videos & reading

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